The fractal bubble model with a cosmological constant

We generalize the fractal bubble model (FB), recently proposed in the literature as an alternative to the standard $\Lambda$CDM cosmology, to include a non-zero cosmological constant. We retain the same volume partition of voids and walls as the original FB model, and the same matching conditions for null geodesics, but do not include effects associated with a nonuniform time flow arising from differences of quasilocal gravitational energy that may arise in the coarse-graining process. The Buchert equations are written and partially integrated and the asymptotic behaviour of the solutions is given. For a universe with $\Lambda=0$, as it is the case in the FB model, an initial void fraction with hyperbolic curvature evolves in such a way that it asymptotically fills completely our particle horizon. Conversely, in presence of a non vanishing $\Lambda$, we show that this does not happen and the voids fill a finite fraction $f_{v_{\infty}}<1$, where the value of $(1-f_{v_{\infty}})$ is expected to depend on $\Lambda$ and the initial fraction $f_{vi}$ and also to be small. For its determination, a numerical integration of the equations is necessary. Finally, an interesting prediction of our model is a formula giving a minimum allowed value of present day dark energy as a function of the age of the universe and of the matter and curvature density parameters at our time.